Electrons in Solids Carbon as Example - PowerPoint PPT Presentation

Electrons in Solids Carbon as Example. Electrons are characterized by quantum numbers which can be measured accurately, despite the uncertainty relation. In a solid these quantum numbers are: Energy: E Momentum: p x,y,z E is related to the translation symmetry in time (t),

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Electrons are characterized by quantum numbers which can be measured accurately, despite the uncertainty relation.

In a solid these quantum numbers are:

Energy: E

Momentum: px,y,z

E is related to the translation symmetry in time (t),

px,y,zto the translation symmetry in space (x,y,z) .

Symmetryin time allows t=E=0(from E·t ≥ h/4)

Symmetry in space allows x=p=0(from p ·x ≥ h/4)

The quantum numbers px,y,zlive in reciprocal spacesince p=ћk.Likewise, the energy Ecorresponds to reciprocal time. Therefore, one needs to think in reciprocal space-time, where large and small are inverted (see Lecture 6 on diffraction).

Analogous to Bohr’s quantization condition one requires that an integer number n of electron wavelengths fits around the circumference of the nanotube. (Otherwise the electron waves would interfere destructively.)

This leads to a discrete number of allowed wavelengths n and k values kn=2/n .(Compare the quantization condition for a quantum well, Lect. 2, Slide 9). Two-dimensional k-space gets transformed into a set of one-dimensional k-lines (see next).

(A) Wrapping vectors (red) and allowed wave vectors kn (purple) for (3,0) zigzag, (3,3) armchair, and (4,2) chiral nanotubes. If the metallic K-point lies on a purple line, the nanotube is metallic, e.g. for (3,0) and (3,3). The (4,2) nanotube does not contain K, so it has a band gap. All armchair nanotubes (n,n) are metallic, since the purple line through  contains the two orange K-points. Note that the purple lines are always parallel to the axis of the nanotube, since the quantization occurs in the perpendicular direction around the circumference.

(B) Band structure of a (6,6) armchair nanotube, including the metallic K-point (orange dot). Each band corresponds to a purple quantization line. Their spacing is Δk = 2/1= 2/circumference= 1/ radius.

D(E) is defined as the number of states per energy interval. Each electron with a distinct wave function counts as a state.

D(E) involves a summation over k, so the k-information is thrown out.

While energy bands can only be determined directly by angle-resolved photoemission, there are many techniques available for determining the density of states.

By going to low dimensions in nanostructures one can enhance the density of states at the edge of a band (E0). Such “van Hove singularities” can trigger interesting pheno-mena, such as superconductivity and magnetism.